Consider the points \( P(1,5,5) \) and \( Q(7,7,11) \). a. Find \( \overrightarrow{P Q} \) and state your answer in two forms: \( \langle a, b, c\rangle \) and \( a i+b j+c k \). b. Find the magnitude of \( \overrightarrow{P Q} \). c. Find two unit vectors parallel to \( \overrightarrow{P Q} \). a. Find \( \overrightarrow{P Q} \). \( \overrightarrow{P Q}=\langle 6,2,6\rangle=(6) i+(2) j+(6) \mathbf{k} \) b. Find the magnitude of \( \overrightarrow{P Q} \). The magnitude of \( \overrightarrow{P Q} \) is \( 2 \sqrt{19} \). (Type an exact answer, using radicals as needed.) c. Find two unit vectors parallel to \( \overrightarrow{P Q} \). The unit vector in the direction of \( \overrightarrow{P Q} \) is \( \langle\square, \square, \square \). (Type exact answers, using radicals as needed.)

To solve the questions about the points \( P(1,5,5) \) and \( Q(7,7,11) \), we start with part a. ### a. We have \[ \overrightarrow{P Q} = Q - P = \langle 7-1, 7-5, 11-5 \rangle = \langle 6, 2, 6 \rangle \] This can also be expressed as \[ 6i + 2j + 6k. \] ### b. To find the magnitude of \( \overrightarrow{P Q} \), we use the formula: \[ |\overrightarrow{P Q}| = \sqrt{6^2 + 2^2 + 6^2} = \sqrt{36 + 4 + 36} = \sqrt{76} = 2\sqrt{19}. \] ### c. A unit vector is found by dividing the vector by its magnitude. The unit vector in the direction of \( \overrightarrow{P Q} \) is \[ \hat{u} = \frac{1}{|\overrightarrow{P Q}|} \overrightarrow{P Q} = \frac{1}{2\sqrt{19}} \langle 6, 2, 6 \rangle = \left\langle \frac{6}{2\sqrt{19}}, \frac{2}{2\sqrt{19}}, \frac{6}{2\sqrt{19}} \right\rangle = \left\langle \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right\rangle. \] For two unit vectors parallel to \( \overrightarrow{P Q} \), we can take the unit vector and its negative: \[ \left\langle \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right\rangle \quad \text{and} \quad \left\langle -\frac{3}{\sqrt{19}}, -\frac{1}{\sqrt{19}}, -\frac{3}{\sqrt{19}} \right\rangle. \]